Enhanced power grid performance through Gorilla Troops Algorithm-guided thyristor controlled series capacitors allocation

This article introduces an innovative application of the Enhanced Gorilla Troops Algorithm (EGTA) in addressing engineering challenges related to the allocation of Thyristor Controlled Series Capacitors (TCSC) in power grids. Drawing inspiration from gorilla group behaviors, EGTA incorporates various methods, such as relocation to new areas, movement towards other gorillas, migration to specific locations, following the silverback, and engaging in competitive interactions for adult females. Enhancements to EGTA involve support for the exploitation and the exploration, respectively, through two additional strategies of periodic Tangent Flight Operator (TFO), and Fitness-based Crossover Strategy (FCS). The paper initially evaluates the effectiveness of EGTA by comparing it to the original GTA using numerical CEC 2017 single-objective benchmarks. Additionally, various recent optimizers are scrutinized. Subsequently, the suitability of the proposed EGTA for the allocation of TCSC apparatuses in transmission power systems is assessed through simulations on two IEEE power grids of 30 and 57 buses, employing various TCSC apparatus quantities. A comprehensive comparison is conducted between EGTA, GTA, and several other prevalent techniques in the literature for all applications. According to the average attained losses, the presented EGTA displays notable reductions in power losses for both the first and second systems when compared to the original GTA. Specifically, for the first system, the proposed EGTA achieves reductions of 1.659 %, 2.545 %, and 4.6 % when optimizing one, two, and three TCSC apparatuses, respectively. Similarly, in the second system, the suggested EGTA achieves reductions of 6.096 %, 7.107 %, and 4.62 %, respectively, when compared to the original GTA's findings considering one, two, and three TCSC apparatuses. The findings underscore the superior effectiveness and efficiency of the proposed EGTA over both the original GTA and several other contemporary systems.


Introduction
A static non-linear programming issue that takes into consideration the electrical elements of massive transmission power grids is called the optimal power flow (OPF).While optimizing vital objectives, the primary purpose of the challenge is to identify the steadystate functioning points of all electric elements accessible to the power systems [1,2].The OPF problem takes into account several individual goals, including entire power losses, the fuel costs for power generation electricity, voltage deviations, polluted emissions, and voltage stability index [3].Furthermore, the issue of OPF requires that a set of operational and physical constraints be accurately met.These constraints include those enforced by devices and network limitations, such as switchable capacitor banks, transmission line capacity limits, bus voltages, transformer taps, active and reactive generators' power, and transformer taps [4].Essentially, in order to obtain other dependent variables such as the voltage magnitude at other buses, and the reactive power of the generators, the control variables such as voltage magnitude at generation buses, the active power of the generators, transformer tap settings, and injected reactive power at capacitor buses of the OPF problem must first be determined [5].However, the development of Flexible Alternating Current Transmission Systems (FACTs) provides several controllable compensators with additional capabilities to enhance the power grids operation [6].
The utilization of FACTs devices in power systems enhances the static security of a power system.Numerous machine learning approaches have been researched to produce assessments that are both quick and accurate enough where a number of sensitivity and optimization techniques have been reported for appropriate placement and sizing in order to guarantee the efficacy of FACTs devices [7].Although there hasn't been much use of FACTs in reactive power settings up until recently, there has been a resurgence of interest in this field of study due to recent advances in science [8].Keeping these things in mind, this study provides up-to-date information on how to enhance the power grids operation by optimally allocating FACTs devices into account.TCSC technology is widely employed in real-world electrical networks as a strong and affordable series FACTS device with outstanding performance that permits accurate reliable power flow control of power lines [9].One of the most economical ways to free up the transmission network's capacity to carry more real power is to implement TCSC devices, which offers series compensating features [10,11].Three series FACTS apparatuses of static synchronous series compensator (SSSC), TCSC, and thyristor controlled phase shifter (TCPS) are being taken into consideration and simulated in enhancing the control in multi-area connected electrical networks in order to minimize area frequency fluctuations and tie-line power [12].The damper controllers were created by combining the Integral of Time multi-plied Squared Error (ITSE) as minimization goal and the Improved Particle Swarm Technique (IPST) as a solution tool.The presented TCSC-AGC outperformed TCPS and SSSC in terms of tie-line transmission powers and vibration dampening at area frequencies.Furthermore, sensitivity testing has been conducted to illustrate the robustness of the TCSC-AGC.This result illustrated the significance and advantageous features of the TCSC over SSSC in transmission networks, indicating its practical application.Academic researchers have recently established a range of traditional and metaheuristic approaches of dealing with OPF [13].Among the traditional approaches are, gradient approaches, sequential unconstrained methodology [14], interior point method [15], linear and nonlinear programming [16,17], Newton-based method [18], and fuzzy linear methods [19].It should be mentioned, though, that these techniques do not produce globally optimal solutions and are not beneficial to large electrical networks.Consequently, in an effort to overcome the shortcomings of earlier M.H. Alqahtani et al. approaches, scientists have worked to develop metaheuristic methodologies [20].Even so, these traditional approaches could get stuck in a local optimum since they are dependent on the initial setup and cannot produce the true optimal outcome.Furthermore, each method cannot easily handle integer and discrete variables, where it requires the modelling of particular OPF variations.Consequently, developing metaheuristic techniques is essential to overcoming the above-described limitations [21].
The power system operation in an economical and voltage-secure manner requires the resolution of complex optimization issues requiring advanced computational techniques.Evolutionary computation is a particular approach that has demonstrated its potential to resolve intricate issues.A myriad of population-based methods such as Equilibrium Optimizer (EO) [22], mayfly optimization algorithm [23], jellyfish optimization algorithm [24], artificial rabbits optimization [25], Teaching-Learning Based Optimization (TLBO) [26,27], moth flame combined with quantum computing technique [28], gradient-based optimization algorithm [29], hybridized TLBO method and artificial bee colony [30], combined Arithmetic Optimization Approach (AOA) and aquila optimization (AQUO) [31], and hybrid optimization algorithm of backtracking search with grey wolf techniques [32] are employed to address the OPF.In addition, in Ref. [33], the Whale optimizer, discussed in Ref. [34], was harnessed to pinpoint optimal placements for FACTS in a power network, while concurrently minimizing operational expenses.This optimization version strategically placed TCSC apparatuses initially on weak lines, identified by the L-index indicating instability.The incorporation of TCSC apparatuses into the transmission system, enhancing corresponding available transfer capability, was also investigated in Ref. [35].In this study, TCSC apparatus locations were predetermined based on the AC power distribution factor (ACPTDF).Subsequently, TLBO was employed to ascertain the pertinent compensation reactance values.Meanwhile, the Gorilla Troops Algorithm (GTA) was previously applied to OPF with TCSC modules in Ref. [36], albeit without due consideration for the allocations of the TCSC.In a distinct approach, a modified crow search technique, detailed in Ref. [37], was employed for OPF with adjustments that amalgamate an innovative bat strategy.Furthermore, the integration of a TCSC apparatus into the electrical system to enhance angle stability, using a PID control system with a filtered component, was explored in Ref. [38].This study introduced a novel combination of sine cosine optimizer and evolutionary programming to derive optimal stabilizer settings, considering diverse operating conditions and the established power system model featuring a single-machine infinite-bus.Additionally, a specialized EO version has been devised for TCSC apparatus allocation in power systems, aiming to minimize voltage deviations, overload, and losses [39].A modified NSGA-III with constraint management, outlined in Ref. [40], was introduced to minimize energy loss, featuring an environmental selecting procedure and reduced selecting tries.Despite the commendable aspects of these research papers in addressing OPF, it's noteworthy that none of the mentioned methodologies have taken into account the critical aspects of TCSC size and allocation.
In recent years, the development and refinement of optimization algorithms have become crucial for solving complex engineering and scientific problems.These advancements are evident across various domains, including control systems, power systems, and global optimization challenges.Several innovative methods have demonstrated significant improvements in achieving high accuracy and faster convergence rates including Information-Exchanged Gaussian AOA (IEG-AOA) [41], hybrid Symbiotic Organisms Searching and Simulated Annealing (hSOS-SA) [42] and modified Salp Swarm Optimizer (mSSO) [43], improved SOS (ISOS) [44], Honey Badger WHO WHO is adopted for designing intelligent fuzzy tilt integral derivative controller Renewable-dominated micro-grid [48] WHO is designed for tuning Fractional-order proportional derivative controller IEEE 39 power system [49] GTA Artificial GTA is adopted for designing Fractional order PID controller Interconnected power system [50] MPA MPA is designed for tuning One plus proportional derivative with filter controller Microgrid system, IEEE 39 test bus validation [51] FO Firefly algorithm for parameter estimation is performed under various noise levels Power system harmonics analysis [52] FoKO A fractional order component and a local escaping strategy are involved in KO algorithm CEC 2017 benchmarks with application for IEEE 33 and 69-nodes distribution systems [53] A FO-KO algorithm is utilized for tuning a (PI-(1+DD) controller for frequency stability control CEC 2020 suite and two-area thermal and wind power system [54,55] CSO Swarm intelligence-based optimization Robustness against different SNRs [59] Power system harmonics, parameter estimation [56] FHGD Fractional hierarchical gradient descent and fractional order adaptation are utilized.
Nonlinear system identification, convergence analysis [57] EFDLMS Enhanced fractional derivative based LMS is introduced while it was effectively applied in Ref. [58] as well.
Recently introduced by Abdollahzadeh et al. [60], the GTA is an innovative optimization approach inspired by the social behaviors of gorilla groups.Emphasizing the hierarchical structure led by a silverback gorilla, GTA replicates key aspects of gorilla life, incorporating exploration strategies like migration to unknown and known locations, and interactions with other gorillas.During the exploitation phase, the algorithm mimics behaviors such as following the silverback and engaging in competitive interactions for adult females [61].GTA is developed based on these observations [62].The objective of this method is to handle optimization issues in an economical manner while demonstrating great performance and adaptability in different fields.In engineering applications, GTA is user-friendly because it has few configurable parameters.Numerous engineering problems have been successfully tackled by it, such as the OPF issue in electricity energy systems [63], frequency control in virtual power plants [64], bidding system for social welfare maximisation in competitive power systems integrating wind farms [19], coordination of directional overload relays in power systems [65], determination of fuel-cell parameter estimation [66], and frequency control of microgrids [67].Not only that, but in Ref. [68], a quantum version of a GTA was presented and combined with a quantum-inspired evolutionary algorithm that drew inspiration from the idea of quantum computing [69].This quantum framework was used to adjust the Tilted-Integral-Derivative controller's settings in a variety of power system stabilizers.It included superposition of quantum states, intervention, and entanglement.To tune PID Controller in micro-robotics apparatuses, a combination method relying on the AOA and GTA was developed in Ref. [70].In order to improve location update efficiency, a hybridized method centered on the Pelican Optimization Approach (POA) and GTA was created in Ref. [71].This algorithm combines the predatory qualities of gorillas with the searching abilities of pelicans.The efficacy of this combination strategy was shown in training a Deep Feedforward Neural Network's weights for the purpose of detecting abnormal activity by humans in smart environments.
The allocation of TCSC in power grids is a critical optimization problem with significant implications for improving the efficiency and stability of power systems.Traditional optimization algorithms, including some metaheuristics, often struggle with issues such as premature convergence and getting trapped in local optima.These limitations hinder their effectiveness in solving high-dimensional, complex problems like TCSC allocation.In order to improve upon the GTA, this research suggests an enhanced GTA, EGTA, by merging it with a Fitness-based Crossover Strategy (FCS) and a periodic Tangent Flight Operator (TFO).The EGTA is developed for solving engineering problems and designed for distributing and sizing the apparatuses in power grids for minimizing the whole losses as a crucial objective task.The EGTA is improved by using FCS for promoting its exploration.Contributing to the exploitation step, which entails a recurrent TFO, also solves it.In engineering disciplines, this optimizer is simple to use and has minimal options to change.The following are the primary contributions that this research cites.• Enhanced Algorithm Development: an innovative EGTA including FCS and TFO is presented addressing common metaheuristic challenges such as premature convergence and local optima entrapment.• Performance Evaluation: The proposed EGTA demonstrates significant improvements against GTA and other state-of-the-art algorithms on benchmarking CEC suite and the TCSC allocation problem.• Statistical Validation: Using the Friedman ANOVA test, statistical significance of the results is validated across 28 different functions, confirming the robustness and superiority of EGTA with highly significant p-values.• Practical Application: The proposed EGTA is applied to optimize the allocation of TCSC in power grids, showing its practical utility and effectiveness in solving real-world engineering problems.The proposed EGTA derives significant robustness metrics compared to GTA and other contemporary methods that is previously reported in the literature.

TCSC modelling
The TCSC, a series-type FACTS component with fast reaction times, excellent performance, and low cost, is currently one of the most often utilized elements in the system.TCSC technology can operate in two reactive modes: inductive and capacitive.Hence, it is possible to modify the reactance of the associated line in either an upward or downward direction.It is connected in series with a line as shown in Fig. 1(a).The TCSC is comprised of an inductance (L) controlled by a valve situated between two thyristors (T1 and T2) and a capacitance (C) arranged in parallel [72].The TCSC, a series-type FACTS component with fast reaction times, The valve's operation is decided by the angle of extinction (α), which can be adjusted to in the range between 90 • and 180 • [73].
As demonstrated in Fig. 1(b), the compensator TCSC injected a variable capacitive reactance (X TCSC ) into the transmission line.The resulting equation determines the controlled thyristors' angle (α).This angle directly impacts the representation of X TCSC [74,75].Thus, the TCSC's reactance gets expressed by the transmission-line reactance (X Line ).The TCSC apparatus's (X TCSC ) essential value can be computed by applying Eq. ( 1) [76,77] in order to stop transmission line overcompensation.
Eq. ( 1) can be rearranged when replacing the terms X L (α) and X C with their values in Eq. ( 2) and Eq.(3) as follows:

Losses minimization and constraints of TCSC allocation
The primary goal is to reduce overall losses in order to technically enhance the electrical system performance and the overall voltage profile.This objective (OBV) can be computationally expressed as follows [78]: The TCSC allocation challenge requires the fulfilment of numerous equality constraints and inequalities constraints to the independent as well as dependent variables.As stated in Eqs. ( 6) and (7), the conditions for reactance compensation, independent variables, and TCSC allocations must be fulfilled.
Regarding independent variables, reactive power infusion from Var sources, tap adjustments, generator voltage, and the generator output electrical power are all subjected to limitations managed by Eqs.8-11.Furthermore, with regards to dependent variables, limitations pertaining to apparent power flow across the transmission lines, bus voltage, and generator electrical reactive power production are handled by Eqs.12-14.

QI min
Accordingly, there must be equality constraints maintained in the reactive as well as active loading balance computations at each bus.The completion of the load flow method fully satisfies these constraints.

Original GTA
The cooperative behaviours of gorillas constitute an inspiration for the GTA algorithm's optimization procedure [60].Three alternative approaches are used to investigate the issue space: travelling in the direction of unknown regions, travelling in the direction of various areas, and travelling along a predetermined path.Two strategies are used in the exploitation stage: monitoring the silverback, which acts as the leader, and competing with adult females.
Every gorilla in the exploration phase is viewed as a possible solution, and the most capable gorilla is named the silverback, in charge of the optimization processes.Before the optimization process is initiated, an attribute called "p" arbitrarily selected between 0 and 1, determines the migration manner to use.The move to an unidentified place is selected if a randomly selected magnitude, called " m 2 ," has a lower than the attribute "p".The practice of approaching nearby gorillas is favoured, though, if " m 2 ″ is more than "p" and surpasses 0.5.Finally, the migration in the direction of a specific site is chosen if "m 4 ″ is larger than "p" yet below 0.5.Equation (15) using can be used to explain the different exploration strategies.
Therefore, the parameters C, Z and L are quantitatively described as: Two techniques of responding to the silverback or competing for females were specifically designed for the GTA algorithm's exploitation phase.Choosing which approach to use is determined by contrasting one parameter, represented by Equation (10), with another, represented by W. Prior to the optimization procedure, a numerical value for the factor W must be supplied.As per reference [60], it has been configured at 0.8.
If the value of C is greater than or equal to W, as expressed in Equation ( 15), the strategy of following the silverback gorilla is activated.At the contrary conjunction, the alternative approach is applied if C is less than W. When gorillas reach adulthood, they compete fiercely with one another for the attention of females.
Equation ( 21) is applied in order to replicate the violent character of the competition by simulating the strong influence on the individuals' dimensions.
The main steps of the original GTA in handling the optimal TCSC allocations in power grids are summarized in the pseudocode 1.As shown, the comparison between C and W in the exploitation portion of the GTA method determines whether to adhere to the silverback or compete for females.When C is larger than or identical to W, the competition approach is engaged; conversely, when C is less than W, the silverback approach is chosen.By introducing dynamics that mimic gorilla population behaviours, these tactics improve the optimization process.Pseudocode 1. GTA steps 1. Set parameters: GN (population size), TM (maximum iterations), β, p, W 2. Insert data of the system under study: Generation data, Lines data and the boundaries of XTCSC(α) k regarding Eq. ( 6) and QIVr, Vg m , Pg m and Tp k regarding Eqs.

Proposed EGTA
This section presents EGTA, an improved version of the GTA optimizer that combines two changes to provide the conventional GTA with more exploratory and exploitation features.A FCS is introduced in the initial modification to support the discovery stage [79,80].Two randomly chosen gorillas are chosen for every dimension in the suggested FCS, and an alternative solution is produced using the lowest fitness metric.For each dimension, two gorillas are randomly chosen which are Ɠ R1 and Ɠ R2 from the population which its size is typically denoted as "GN".Next, the fitness values of these two randomly chosen gorillas, Fit(Ɠ R1 ) and Fit(Ɠ R2 ), are obtained.These fitness values represent the performance or quality of the solutions provided by the gorillas in the current population.The FCS method then compares the fitness values of Ɠ R1 and Ɠ R2 to determine which gorilla has the minimum fitness.The minimum fitness indicates the better-performing solution.Based on the minimum fitness criterion, the FCS method selects an assistant solution vector, denoted as Ɠd, as follows: In other words, the position of the gorilla with the better fitness becomes the assistant solution vector.Based on that FCS, a new gorilla position can be generated as follows: The present location of the gorilla is replaced with a newly produced seeking vector by utilizing the FCS's suggested method.As a result, the FCS keeps the computation pace constant by maintaining an equivalent amount of function executions.The gorillas' existing locations will be replaced by a set of newly created locations that are produced by repeating this method in every dimension.The FCS technique adds diversity to the discovery process and promotes exploration by choosing two randomly chosen gorillas and assigning the new place to the gorilla with the lowest fitness.

M.H. Alqahtani et al.
In addition, a periodic tangent function [81] is incorporated for use in the process of exploitation by adding a TFO in the following manner: This approach allows for an efficient investigation of searching area.The TFO is added to Eq. ( 19) by the suggested EGTA approach.By reducing the distance between the gorilla and the silverback, this modification significantly lowers the end step size and raises the objective rating.The following is a mathematical description of this model: The key steps of the proposed EGTA in handling the optimal TCSC allocations in power grids are summarized in the pseudocode 2. This provided pseudocode effectively encapsulates the core concepts, with clear steps for initialization, fitness evaluation, and iterative optimization.The incorporation of TFO to improve convergence and precision is a notable enhancement.It can be summarized in four main steps.
Step 1. Initialization: It includes parameter setup, system data input and creation of the initial population of gorillas with random positions within specified bounds.Also, the fitness score for each solution is evaluated.
Step 2. Main Loop for Iterations: It involves parameters update (C, Z, and L) and the application of exploration and exploitation techniques.Also, the fitness score for each new generated solution is evaluated.In this step, the exploitation is activated if the new position does not provide better fitness by selecting two random gorillas and generating a new position based on the one with lower fitness using FCS.
Step 3. Follow or Compete: If the parameter C is greater than or equal to W, the gorilla's position is updated by following the best solution found so far.On the other side, if C is less than W, the TFO is applied to refine the position updates, promoting more precise exploration and improved convergence towards the optimal solution.
Step 4. Final Updates and Return: It includes the fitness and best solution update and returning the optimal solution.

Results and discussion
In this section, we delve into the application of the proposed novel EGTA in two distinct dimensions.Firstly, we conduct simulations on benchmark functions, specifically addressing the CEC 2017 benchmarks.These simulations involve a thorough comparison with several recently developed metaheuristic algorithms.Secondly, simulations are extended to tackle TCSC allocation challenges within power networks, focusing on two IEEE original power systems comprising 30 and 57 buses.

Evaluation of application performance using CEC 2017 benchmark models
Ascertaining the effectiveness of optimization strategies can be challenging without a formal proof of their efficacy.Benchmark functions play a crucial role in evaluating the performance of these strategies.In this context, we assess the performance of the proposed EGTA and GTA techniques through the utilization of the CEC 2017 competition as a benchmark [82].This competition encompasses a variety of routines designed to evaluate different attributes, including unimodal, multimodal, mixed, and composite  functions.An overview of the unrestricted benchmarking functions employed in our analysis is described in Ref. [73].Across all 28 benchmarking functions, we consider a dimension of 30 control variables, with their bounds set at [− 100, 100].The proposed EGTA undergoes evaluation in comparison to the conventional GTA, considering CEC 2017 single-objective optimization.The mean convergence characteristics of EGTA and GTA are visually presented in Fig. 2, providing insights into their performance on CEC 2017 problems.The mean convergence is utilized rather than the best run to provide a more representative and statistically robust comparison of the algorithms' performance.Furthermore, a comprehensive comparison involves several contemporary optimization techniques, namely AQUO [83], red kite optimization (RKO) [84], and subtraction-average-based algorithm Technique (SAT) [85].The specific settings and a range of successful applications for each contrasted technique are detailed in Table 2. From this table, the number of fitness function evaluations for all algorithms are similar by 15,000 in order to conduct a fair comparison.In this regard, the number of gorillas solutions is fifteen since GTA and EGTA perform two function evaluations per solution which are double the times of the others.Metaheuristic techniques are inherently stochastic, as they rely on generating several random parameters.Because of this stochastic nature, different runs of the same algorithm can start from different initial points, and this variability is a fundamental characteristic of these methods.To ensure a fair and comprehensive comparison, each algorithm was executed independently for 50 runs.The performance was then assessed using statistical measures such as the maximum, mean, minimum, and standard deviation of the results across these runs.Table 2 furnishes statistical metrics, encompassing the best, mean, worst, and standard deviation (Std) outcomes for the compared methods.
Remarkably, as illustrated in Table 3, the EGTA technique exhibits superior efficacy, consistently achieving the lowest statistical indices across the majority of benchmark functions.Moreover, Table 3 provides insights into the outcomes of a Friedman ranking test conducted on the compared algorithms.As evident, the meticulously designed EGTA attains a commendable average rank of 1.848,  securing the topmost position.In the subsequent tier, DMO achieves a mean rank of 2.8303, while the original GTA occupies the third spot with a rank of 3.424.Following suit, RKO and SAT claim the fourth and fifth positions, boasting mean ranks of 3.634 and 4.495, respectively.Notably, AQUO lingers at the bottom with a mean rank of 4.767.These results underscore the remarkable performance of EGTA, showcasing improvement reductions of 34.7 %, 46.02 %, 49.14 %, 58.88 %, and 61.2 % compared to DMO, GTA, RKO, SAT, and AQUO, respectively.Such compelling advancements highlight EGTA's efficacy and robustness across diverse optimization scenarios.
Using Friedman ANOVA Test, Table 4 provides    extraordinarily low, with many on the order of 10E− 30 or even lower.This indicates highly significant differences between the compared algorithms for most of the functions.Also, the p-values range from a minimum of 1.4445E− 12 to a maximum of 4.4365E− 48.This wide range still falls well below the typical significance threshold (e.g., 0.05), reaffirming that all comparisons reveal significant differences.The mean p-value is 5.1896E− 14, which further emphasizes the overall significant differences across the board.
In order to show the computational complexity analysis for the compared methods, the "Big O analysis" [87] is implemented.Table 5 compares the computational complexity and average runtime of various algorithms (EGTA, GTA, DMO, SAT, RKO, AQUO) across 28 benchmark functions with a dimension of 10.The results indicate that all algorithms have a computational complexity of O (150,000), implying similar theoretical performance in terms of computational steps.However, the average runtime varies significantly.EGTA consistently shows efficient runtimes, typically around 0.3-0.4s, demonstrating a balance between computational effort and efficiency.GTA performs similarly but slightly less efficiently than EGTA.In contrast, DMO exhibits significantly higher runtimes, often exceeding 0.5 s, making it less suitable for applications requiring quick solutions.SAT consistently has the lowest runtimes, often below 0.2 s, but this comes at the cost of solution quality as previously demonstrated.RKO and AQUO show moderate performance, with runtimes generally between 0.2 and 0.6 s, but still lag behind EGTA.Therefore, EGTA proves to be efficient and reliable, making it a preferable choice for practical applications needing quick and high-quality solutions.

TCSC installations for IEEE 30-bus grid
In this part, the optimal TCSC allocations are managed through the IEEE standard 30-bus system, as can be seen in Fig. 3.There are 4 transformers, 41 lines, 30 nodes, and 9 compensators in this system [88,89].The tap positions are 0.90 p.u., whilst the maximum generating voltage is 1.10 p.u.Moreover, the voltage limitations for the load buses are between 1.05 and 0.95 p.u., whilst the generator bus has limits of 1.10 and 0.90 p.u.
The proposed EGTA is compared to the original GTA as well as other modern algorithms previously applied in the literature which are AQUO [73], DMO [73], Improved DMO (IDMO) [73], Artificial Ecosystem Optimization (AEO) [91], SAT [91], Improved SAT (ISAT) [91] and Grey Wolf Optimization (GWO) [91].The proposed novel EGTA and the original GTA are run 20 times independently, with 300 iterations and 50 searching individuals.Three distinct cases are examined, taking into account one, two, and three TCSC apparatuses, subject to the number of candidate apparatuses provided.

Case 1: one TCSC apparatus
In this specific instance, the innovative designed EGTA is applied for the optimization of a single TCSC device allocation, with a comprehensive comparison against the original GTA, SAT, ISAT, AEO, AQUO, and GWO detailed in Table 6.This tabulation presents essential details regarding the TCSC apparatus, encompassing location, size, and crucial control variables like generator voltage, output power, tap value, and VAr source inclusion power.Furthermore, Fig. 4 provides visual representations of the convergence trajectories for both the presented EGTA and the original GTA.Notably, the proposed EGTA exhibits outstanding performance, achieving a minimal power loss of 2.805486 MW.The algorithm identifies the transmission line  as the optimal TCSC placement, featuring a 50 % reduction in size from the installed line reactance.Remarkably, compared to the baseline scenario, the novel EGTA secures an impressive 48.10 % reduction in power losses.Moreover, the EGTA demonstrates a notable 1.01 % decrease in power losses compared to the original GTA.Additionally, when juxtaposed with AEO, SAT, ISAT, AQUO, IDMO, DMO, and GWO, the proposed EGTA showcases reductions in power losses by 1.35 %, 8.347 %, 0.575 %, 6.171 %, 0.361 %, 7.072 %, and 7.562 %, respectively.These compelling results underscore the efficiency and superiority of the EGTA in optimizing TCSC allocation, showcasing its potential for significant advancements in power grid management.
Additionally, a comprehensive statistical assessment is conducted, and the box plot illustrating the results of both the conventional GTA and the newly proposed EGTA for Case 1 is depicted in Fig. 5.The superiority of the proposed EGTA becomes evident as it accumulates the fewest metrics from the achieved objective values.Regarding mean losses, the original GTA records losses of 2.865 MW, whereas the suggested EGTA achieves the lowest losses at 2.817 MW, showcasing an improvement of 1.6593 % over the GTA.In terms of worst-case losses, the original GTA reports losses of 2.887 MW, while the suggested EGTA attains the lowest losses at 2.847 MW,   representing a 1.4 % improvement over the GTA.When considering standard deviation in losses, the original GTA exhibits losses of 0.01388 MW, whereas the suggested EGTA demonstrates the lowest losses at 0.0112 MW, indicating a notable 19.07 % improvement over the GTA.These findings underscore the consistent and superior performance of the proposed EGTA in optimizing TCSC allocation for enhanced power grid management.

Case 2: two TCSC apparatus allocations
In this case, the proposed EGTA is applied to optimize the allocation of two TCSC devices.Against the backdrop of the GTA, SAT, ISAT, AEO, AQUO, and GWO, the outcomes detailing the location, size of the TCSC devices, and the most suitable control variables-output power, generator's voltage, tap value, and Var source's inclusion power-are presented in Table 7. Furthermore, Fig. 6 illustrates the convergence curves for both the EGTA and the original GTA.Evidently, the suggested EGTA achieves the minimal power loss of 2.784025 MW.The proposed EGTA designates transmission lines (4)(5)(6)(7)(8)(9)(10)(11)(12) and  as the optimal locations for the TCSC, incorporating a 50 % addition and 50 % subtraction, respectively, from the installed line reactance.Compared to the initial scenario, the EGTA brings about a substantial 47.73 % reduction in power losses.The suggested EGTA demonstrates a noteworthy decline of 0.18 % in power losses in contrast to the original GTA.Additionally, in comparison to AEO, SAT, ISAT, AQUO, IDMO, DMO, GWO, the proposed EGTA exhibits reductions in power losses by 2.98 %, 11.42 %, 1.29 %, 7.57 %, 0.67 %, 7.98 %, 15.91 %, respectively.These results underscore the efficacy of the proposed EGTA in optimizing dual TCSC allocation for enhanced power grid efficiency.In this instance, a statistical assessment is conducted, and the box plot portraying the results of both the original GTA and the innovative EGTA is depicted in Fig. 7. It's apparent that the EGTA attains superior outcomes by accumulating the fewest indices from the obtained objective values.Concerning mean incurred losses, the original GTA records losses of 2.802 MW, while the proposed EGTA identifies the lowest mean losses at 2.875 MW.For the worst incurred losses, the proposed EGTA registers the least losses at 2.825 MW, while the GTA incurs losses of 3.130 MW.Additionally, the proposed EGTA identifies the minimum standard deviation of losses at 0.0692 MW, compared to the GTA's standard deviation losses of 0.0146 MW.Against the GTA, the proposed EGTA demonstrates enhancements of 2.545 %, 9.746 %, and 78.904 % for mean, worst, and standard deviation of incurred losses, respectively.

Case 3: three TCSC apparatus allocations
In this specific scenario, three TCSC units are examined to minimize power losses.Table 8 outlines the TCSC locations, sizes, and optimal control variables-output power, generator's voltage, tap value, and Var source's inclusion power-across EGTA, GTA, SAT, ISAT, AEO, AQUO, and GWO.Additionally, Fig. 8 illustrates the convergence trends for EGTA and the original GTA.Notably, EGTA achieves the least power loss at 2.759085 MW, selecting optimal locations (4-12), , and (2-5) with specific size adjustments.The proposed EGTA attains a substantial 47.31 % reduction in power losses compared to the initial configuration, emphasizing its efficacy in optimizing TCSC allocation for enhanced power grid performance.For this case, the proposed EGTA records an improvement of 1.618 % compared to the original GTA which achieves losses of 2.804 MW.The suggested EGTA achieves a noteworthy decrease of 1.65 % in the power losses when compared to the original GTA.In addition, compared to the AEO, SAT, ISAT, AQUO, IDMO, DMO, GWO, the proposed EGTA reduces power losses by 4.38 %, 10.04 %, 2.24 %, 7.61 %, 1.29 %, 9.35 %, 15.51 %, respectively.
In a statistical assessment, the box plot in Fig. 9 illustrates outcomes for both the original GTA and the proposed EGTA in Case 3. The EGTA excels with minimal indices, showcasing superior results.For mean losses, the original GTA records 2.876 MW, while EGTA achieves the lowest at 2.784 MW.Compared to GTA, EGTA consistently outperforms, attaining the least losses at 2.811 MW for worstcase scenarios, while GTA registers 3.079 MW in losses.This highlights EGTA's enhanced performance, showcasing improved reductions in worst-case losses against GTA's outcomes.

Table 6
Optimized controller parameters and regarding outcomes of EGTA versus GTA and other algorithms for TCSC allocations for Case 1 (IEEE 30-bus system).

Voltage enhancement with EGTA-Driven TCSC installations in IEEE 30-bus system
The voltage profiles of grid buses resulting from the application of the proposed EGTA and the integration of TCSC in three distinct cases are illustrated in Fig. 10, providing a comparative analysis with the initial case.Notably, the voltage levels at various grid buses showcase significant improvements across these scenarios.The most pronounced enhancement is observed at the terminal grid bus (No. 30), where noteworthy improvements of 15.77 %, 15.75 %, and 15.79 % are realized for Cases 1, 2, and 3, respectively, marking a transition from 0.9012 per unit (p.u.) to 1.0699, 1.0697, and 1.0702 p.u. Averaging across the entire system, the mean voltage value of 0.9842 p.u. in the initial scenario undergoes a considerable 9.5 %, 9.49 %, and 9.51 % enhancement with the incorporation of one, two, and three TCSC installations, respectively.

TCSC installations for IEEE 57-bus grid
This part uses the standard IEEE 57-bus transmission network.There are 57 nodes, 80 lines, 7 generators, 17 on-load tap changing transformers, and 3 capacitive sources on buses in this system [92].In order to achieve minimum power losses, the three situations under study are examined with consideration for one, two, and three TCSC apparatuses.Table 9 tabulates their obtained control variables, the EGTA and GTA are applied.Alternatively, Fig. 11 reveals the regarding converging features of the proposed EGTA and the original GTA.
As shown in Table 9, for the first case, the proposed EGTA identifies the transmission line  as the optimal TCSC placement, featuring a 49.64 % reduction in size from the installed line reactance.Also, it sets the other control variables simultaneously which reduces the power losses from 27.835 MW in the initial case to 9.3052 MW while the GTA achieves counterpart losses of 9.347 MW.Remarkably, compared to the baseline scenario, the developed EGTA secures an impressive 66.55 % reduction in power losses.Moreover, the EGTA demonstrates a notable 0.449 % decrease in power losses compared to the original GTA.
In the second scenario, the EGTA optimization identifies the optimal TCSC placements at transmission lines (9-55) and , resulting in a significant 17.68 % and 50 % reduction in size from the installed line reactance.Additionally, it achieves a reduction in power losses to 9.119 MW, surpassing the GTA's losses of 9.362 MW.Notably, the developed EGTA showcases a substantial 2.6 %

Table 7
Optimized controller parameters and regarding outcomes of EGTA versus GTA and other algorithms for TCSC allocations for Case 2 (IEEE 30-bus system).
M.H. Alqahtani et al. decrease in power losses compared to the original GTA.
Moreover, Fig. 12 presents a box plot illustrating the outcomes of the proposed EGTA and the original GTA.It is evident that the EGTA outperforms by accumulating fewer indices among the obtained objective values.The implications from this figure are as follows.
• In the first scenario, the proposed EGTA identifies the lowest losses at 9.4601 MW based on mean acquired losses, while the original GTA records mean losses of 10.0743 MW.Thus, in comparison to the original GTA results, the proposed EGTA exhibits improvement reductions of 6.096 %. • In the second scenario, the proposed EGTA identifies the lowest losses at 9.368 MW based on mean acquired losses, while the original GTA records mean losses of 10.0844 MW.Consequently, in comparison to the original GTA results, the proposed EGTA demonstrates improvement reductions of 7.107 %. • In the third scenario, the proposed EGTA identifies the lowest losses at 9.3292 MW based on mean acquired losses, while the original GTA records mean losses of 9.781 MW.Thus, in comparison to the original GTA results, the proposed EGTA displays improvement reductions of 4.62 %.
For the IEEE-57 bus system, the voltage profiles of grid buses resulting from the application of the proposed EGTA and the integration of TCSC in three distinct cases are illustrated in Fig. 13, providing a comparative analysis with the initial case.Notably, the voltage levels at various grid buses showcase significant improvements across these scenarios.The most pronounced enhancement is observed at the terminal grid bus (No. 31), where noteworthy improvements of 5.58 %, 8.42 %, and 7.52 % are realized for Cases 1, 2, and 3, respectively, marking a transition from 0.9353 per unit (p.u.) to 0.9912, 1.0219, and 1.012 p.u.
Therefore, the application of more TCSC devices in this system with the EGTA implementation helps in achieving better voltage profile.Specifically, in terms of average voltages across the entire system, the initial mean value of 0.9942 p.u. experiences a        substantial 4.15 %, 5.79 %, and 6.65 % enhancement with the introduction of one, two, and three TCSC installations, respectively.This results in a transition from 0.994217 per unit (p.u.) to 1.03725, 1.05536, and 1.064993 p.u., signifying a noteworthy improvement in voltage levels.Table 10 presents a comparative analysis of power losses (MW) for different algorithms applied to the IEEE 57-bus system.It compares the performance of the proposed EGTA and the original GTA against two published methods: DMO and IDMO from Ref. [73].In Case 1, the power losses for DMO and IDMO are 13.30243MW and 9.846252 MW, respectively.The GTA reduces these losses to 9.347426 MW.However, the EGTA further reduces the losses to 9.305192 MW, achieving the lowest power loss among the compared algorithms.For Case 2, the power losses recorded by DMO and IDMO are 13.00813MW and 9.951942 MW, respectively.The GTA achieves a lower loss of 9.362048 MW, but the EGTA outperforms all other methods with a further reduced loss of 9.118985 MW.In Case 3, the power losses for DMO and IDMO are 13.20578MW and 9.746247 MW, respectively.GTA reduces the losses to 9.146321 MW, and once again, EGTA achieves the lowest power losses with 9.051817 MW.This consistent reduction across all cases underscores the effectiveness of the EGTA in optimizing TCSC allocation.The incremental improvements from GTA to EGTA demonstrate the significant benefits brought by the enhancements introduced in the EGTA.These comparisons validate the improvements made in the EGTA, demonstrating its superior capability in optimizing TCSC allocation and improving power grid performance.

Conclusion
This study introduces the Enhanced Gorilla Troops Algorithm (EGTA), an advanced metaheuristic designed to address the optimization challenges associated with Thyristor Controlled Series Capacitors (TCSC) allocation in power grids.By enhancing the standard Gorilla Troops Algorithm (GTA) with adaptive mechanisms, EGTA effectively mitigates common metaheuristic issues such as premature convergence and local optima entrapment.Extensive benchmarking on the CEC suite and the TCSC allocation problem demonstrates that EGTA significantly outperforms GTA and other state-of-the-art algorithms in terms of convergence speed and solution quality.The statistical validation of our results using the Friedman ANOVA test further confirms the robustness and effectiveness of EGTA, with highly significant p-values across 28 different functions.The practical application of EGTA to optimize TCSC allocation highlights its utility in real-world engineering problems, showcasing its potential to enhance the efficiency and stability of power systems.This study not only advances the theoretical understanding of metaheuristic optimization but also provides a robust and efficient tool for practical applications.

Advantages and disadvantages
The EGTA offers several advantages over existing optimization algorithms.Firstly, it demonstrates improved optimization performance, outperforming the standard GTA and other state-of-the-art methods in terms of convergence speed and solution quality.This makes it highly effective for tackling complex optimization problems.Additionally, EGTA incorporates adaptive mechanisms that  strike a balance between exploration and exploitation, reducing the risk of premature convergence and enhancing its ability to find global optima.The algorithm's robustness is supported by statistical validation, showing significant differences in performance.Furthermore, EGTA has proven its practical applicability by successfully optimizing TCSC in power grids, highlighting its potential to enhance power system efficiency and stability.Its flexibility allows for handling multi-objective optimization and adapting to dynamic scenarios, making it a versatile tool for various applications.However, there are certain disadvantages to consider.The computational complexity of EGTA can be higher compared to simpler algorithms due to its adaptive mechanisms and enhanced features, leading to longer computation times for large-scale problems.Parameter sensitivity is another challenge, as finding optimal settings may require additional tuning and experimentation.Implementing EGTA may also be more complex, necessitating a deeper understanding of the algorithm's inner workings.Lastly, the effectiveness of EGTA can depend on the specific problem characteristics, which may require problem-specific adjustments for optimal performance.

Future research directions
Future research will focus on further enhancing the performance of EGTA and expanding its applicability within the power system field.Applying EGTA to dynamic optimization problems in power systems can be extended, such as real-time load balancing, dynamic pricing, and adaptive network reconfiguration.Also, extending EGTA to handle multi-objective optimization problems in power systems can be handled, including the simultaneous optimization of power loss reduction, voltage stability, and cost minimization.Additionally, utilizing EGTA to optimize the integration of renewable energy sources into power grids can be performed, addressing challenges such as optimal placement and sizing of distributed generation units, and managing the variability and uncertainty of renewable energy production.Moreover, exploring the application of EGTA in smart grid environments, focusing on optimizing demand response programs, enhancing grid resilience, and improving the efficiency of distributed energy resources management.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
the P-values of the Comparisons of EGTA, GTA, DMO, SAT, RKO and AQUO.The provided p-values indicate the results of the comparisons across 28 different functions.As shown, the majority of the p-values are

Table 1
Recent applied algorithms, its key modifications and validation frameworks.

Table 2
Compared algorithms: Parameters and applications.

Table 3
Statistical outcomes of the EGTA compared to GTA, DMO, SAT, RKO and AQUO for CEC 2017 tasks.

Table 4
Friedman ANOVA test for the 28 functions.

Table 5
Computational complexity and runtime of the compare algorithms for CEC 2017 tasks.

Table 8
Optimized controller parameters and regarding outcomes of EGTA versus GTA and other algorithms for Case 3 (IEEE 30-bus system).Positive or negative signs reveal an increase or decrease in the TCSC-connected transmission line reactance.

Table 9
Optimized controller parameters and regarding outcomes of EGTA versus GTA for TCSC apparatus allocations concerning Cases 1-3 (IEEE 57-bus system).Positive or negative signs reveal an increase or decrease in the TCSC-connected transmission line reactance.